Theorem rnun 5196

Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
rnun	|- ran ( A u. B ) = ( ran A u. ran B )

Proof of Theorem rnun

Step	Hyp	Ref	Expression
1		cnvun 5193	. . . 4 |- `' ( A u. B ) = ( `' A u. `' B )
2	1	dmeqi 4982	. . 3 |- dom `' ( A u. B ) = dom ( `' A u. `' B )
3		dmun 4988	. . 3 |- dom ( `' A u. `' B ) = ( dom `' A u. dom `' B )
4	2, 3	eqtri 2259	. 2 |- dom `' ( A u. B ) = ( dom `' A u. dom `' B )
5		df-rn 4785	. 2 |- ran ( A u. B ) = dom `' ( A u. B )
6		df-rn 4785	. . 3 |- ran A = dom `' A
7		df-rn 4785	. . 3 |- ran B = dom `' B
8	6, 7	uneq12i 3381	. 2 |- ( ran A u. ran B ) = ( dom `' A u. dom `' B )
9	4, 5, 8	3eqtr4i 2269	1 |- ran ( A u. B ) = ( ran A u. ran B )

This proof depends on syntax axioms:   = wceq 1402   u. cun 3218   `'ccnv 4773   dom cdm 4774   ran crn 4775

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-cnv 4782  df-dm 4784  df-rn 4785

This theorem is used by:  imaundi  5200  imaundir  5201  rnpropg  5267  fun  5561  foun  5658  fpr  5897  fprg  5898  sbthlemi6  7279  exmidfodomrlemim  7553